MAC 2103

Module 4

Vectors in 2-Space and 3-Space I

2Rev.F09

Learning Objectives

In this module, we apply our earlier ideas specifically to vectors in 2-space, ℜ2, (in the xy-plane) in two dimensions and to vectors in 3-space, ℜ3,(in the xyz-space) in three dimensions.

http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

3Rev.F09

Learning Objectives (Cont.)

Upon completing this module, you should be able to:

1. Determine the components of a vector in ℜ2 and ℜ3.

2. Perform vector addition, subtraction, and scalar multiplication in ℜ2 and ℜ3.

3. Find the norm of a vector and the distance between points in ℜ2 and ℜ3.

4. Find the dot product of two vectors in ℜ2 and ℜ3.

5. Use the dot product to find the angle between two vectors in ℜ2 and ℜ3.

6. Find the projection of a vector onto another vector in ℜ2 and ℜ3, and express the original vector as a sum of two orthogonal vectors.

7. Find the distance between a point and a line in ℜ2 and ℜ3.

4Rev.09

Vectors in ℜ2 and ℜ3

Introduction to Vectors (Geometric)Introduction to Vectors (Geometric)

Norm of a Vector; Vector OperationsNorm of a Vector; Vector Operations

Dot Product; ProjectionsDot Product; Projections

There are three major topics in this module:

5Rev.F09

What are Vectors in ℜ2 and ℜ3?

• Vectors can be represented as directed line segments or arrows in ℜ2 and ℜ3. • The direction of the arrow specifies the direction of the vector. • A vector that starts from an initial point A and terminates at a point B can be represented as .• A vector is usually denoted in lowercase boldface type (like v) in the textbook or with an arrow above it when we write it by hand. For example: A B

ABu ruu

v =rv=AB

u ruu.

rv⏐ →⏐−rv← ⏐⏐

6Rev.F09

What are Vectors in ℜ2 and ℜ3? (Cont.)

• The magnitude of the vector is the length of the vector.• The vector of length zero is called the zero vector.• Vectors with the same magnitude and same direction are equal to each other. • A vector v in standard position has its starting point at the origin. The coordinates (v1, v2) of the terminal point of v are called the components of v.

v =rv=(v1,v2 )

Note: The negative of vector v is defined to be the vector that has the same magnitude as v but is oppositely directed.

7Rev.F09

If s is any scalar, then a vector of the form sv is called a scalar multiple of v.

For example, if v = (2,-7) and s =- 5, then

sv =srv=s(v1,v2 ) =(sv1,sv2 )

8Rev.F09

• If v and u are any two vectors in standard position, then the sum and difference of the two vectors is also a vector. It’s also a vector in standard position.

v + u=rv+

ru=(v1,v2 ) + (u1,u2 ) =(v1 +u1,v2 +u2 )

v−u=rv−

ru=(v1,v2 )−(u1,u2 ) =(v1 −u1,v2 −u2 )

9Rev.F09

What are the Components of a Vector in ℜ3?

If the initial point of is A(x1,y1,z1) and the terminal point of is B(x2,y2,z2) in ℜ3, then the components of can be obtained by subtracting the coordinates of the initial point from the coordinates of the terminal point.

Example: Suppose the initial point of is A(1,-2,5) and terminal point is B(-1,4,9), then the components of the vector . We see that the vector is equal to the vector v in standard position.

ABu ruu r

v⏐ →⏐

ABu ruu

v =rv=AB

u ruu=(−2,6,4)

ABu ruu

Rev.F09

Example

Suppose

Find the components of

Note: In chapter 1, we would represent these vectors as column matrices:

ru =(−5,1,6)rv=(1,0,−8)

7ru −2

ru+ (−2)

rv=7(−5,1,6) + (−2)(1,0,−8)

=((7)(−5) + (−2)(1),(7)(1) + (−2)(0),(7)(6) + (−2)(−8))=(−37,7,58)

7ru −2

−516

⎢⎢⎢

⎥⎥⎥= −5 1 6⎡⎣ ⎤⎦

10−8

⎢⎢⎢

⎥⎥⎥

= 1 0 −8⎡⎣ ⎤⎦T

Rev.F09

Some Important Properties of a Vector Space

If u, v, and w are vectors in ℜ2, ℜ3, or any vector space and k and s are scalars, then the following hold:

• u + v = v + u b) (u + v) + w = u + (v + w)

c) u + 0 = 0 + u = u d) u + (-u) = 0

e) k(su)= ks(u) f) k(u + v)= ku + kv

g) (k + s) u = ku + sv h) 1u = u

Rev.F09

What is the Norm of a Vector in ℜ3?

• The norm of a vector u, , is the length or the magnitude of the vector u.

• If u = (u1, u2, u3) = (-1, 4, -8), then the norm of the vector u is

• This is just the distance of the terminal point to the origin for u in standard position.

Note: If u is any nonzero vector, then

is a unit vector. A unit vector is a vector of norm 1.

ru = u

ru = u1

2 +u22 +u3

2 = (−1)2 + 42 + (−8)2 =9

Rev.F09

How to Find the Distance Between Two Points?

• If A(x1,y1,z1) and B(x2,y2,z2) are two points in ℜ3, then the distance between the two points is the length, the magnitude, and the norm of the vector .

d = AB

u ruu= (x2 −x1)

2 + (y2 −y1)2 + (z2 −z1)

ABu ruu

Rev.F09

How to Find Dot Product of Two Vectors in Terms of the Components of the Vectors?

If u = (u1, u2, u3) and v = (v1, v2, v3) , then the dot product of the two vectors in terms of the components of the vectors is:

Example: If u = (3, 0, -1) and v = (2, 9, -2) , then the dot product of the two vectors is:

u⋅v =ru⋅

rv=(3)(2) + (0)(9) + (−1)(−2) =

30−1

⎢⎢⎢

⎥⎥⎥

2 9 −2⎡⎣ ⎤⎦

=rurvT =6 + 0 + 2 =8

Rev.F09

How to Find the Angle Between Vectors?

By definition, if u and v are nonzero vectors in ℜ2 and ℜ3 and is the angle between u and v, then the dot product of the two vectors is:

Thus, if u and v are nonzero vectors, the angle can be obtained by:

Note: From the previous slide,

u⋅v =ru⋅

rv cos(θ)

cos(θ) =

Rev.F09

Some Important Properties of theDot Product

If u, v, and w are vectors in ℜ2 and ℜ3 and s is a scalar, then the following relationships hold:

• u · v = v · u

• u · (v + w) = u · v + u · w

c) s (u · v) = (s u ) · v = u · (s v )

d) and

e) if and only if v ≠ 0, and v · v = 0 iff v = 0

If the vectors u and v are nonzero and θ is the angle between them, then θ = π/2 if and only if u·v = 0. Then,

u and v are perpendicular or orthogonal.

v ⋅v =rv⋅

rv = v⋅v

v ⋅v =rv⋅

rv 2 > 0

Rev.F09

How to Find the Projection of a Vector onto Another Vector?

If u and v are vectors in in ℜ2 and ℜ3 and if a ≠ 0, then

(vector component of u along a)

(vector component of u

orthogonal or perpendicular to a)

Thus, the projau and u - projau are orthogonal vectors whose sum is u. The dot product of projau and u - projau is zero.

projau=u⋅aa 2 a=u⋅

u−projau=u−u⋅aa 2 a

(projau)⋅(u−projau) =u⋅aa 2 a

⎝⎜

⎠⎟⋅ u−

u⋅aa 2 a

⎝⎜

⎠⎟ =

(u⋅a)2

a 2 −(u⋅a)2 (a⋅a)

a 2 a 2 =(u⋅a)2

a 2 −(u⋅a)2

a 2 =0

Rev.F09

How to Find the Projection of a Vector onto Another Vector and Express the Original Vector as

the Sum of Two Orthogonal Vectors?

Example

Let u = (3,1,-7) and a = (1,0,5). Find the vector component of u along a and the vector component of u orthogonal to a.

Solution:

Step 1: Find the dot product of the two vectors.

Step 2: Find the norm of a.

u⋅a=ru⋅

ra=(3)(1) + (1)(0) + (−7)(5) =3+ 0 + (−35) =−32

a =ra = (1)2 + (0)2 + (5)2 = 1+ 0 + 25 = 26

Rev.F09

How to Find the Projection of a Vector onto Another Vector? (Cont.)

Step 3: Solve for the vector component of u along a.

Step 4: Solve for the vector component of u orthogonal to a.

projau=u⋅aa 2 a=

−3226

a=−1613

(1,0,5) =(−1613

,0,−8013

u−projau=(3,1,−7)−(−1613

,0,−8013

) =(5513

,1,−1113

(u−projau) + projau=(5513

,1,−1113

) + (−1613

,0,−8013

=(3913

,1,−9113

) =(3,1,−7) =u

Rev.F09

How to Find the Projection of a Vector onto Another Vector? (Cont.)

Step 5: Check to see if the two component vectors are orthogonal.

projau=u⋅aa 2 a=

−3226

a=−1613

(1,0,5) =(−1613

,0,−8013

u−projau=(3,1,−7)−(−1613

,0,−8013

) =(5513

,1,−1113

(projau)⋅(u−projau) =(−1613

,0,−8013

)⋅(5513

,1,−1113

=−880169

+ 0 +880169

Rev.F09

How to Find the Distance Between a Point and a Line?

Example

Find the distance D from the point (-3,1) to the line 4x+3y+4=0.

Solution:

We can use the distance formula in Equation (13)

to find the distance D. In our problem, x0=-3, y0=1, a=4, b=3, and c=4.

D =ax0+by0 + c

a2 +b2

D =ax0+by0 + c

a2 +b2=

(4)(−3) + (3)(1) + 4

42 + 32=

Rev.F09

What have we learned?

We have learned to:

1. Determine the components of a vector in ℜ2 and ℜ3.

2. Perform vector addition, subtraction, and scalar multiplication in ℜ2 and ℜ3.

3. Find the norm of a vector and the distance between points in ℜ2 and ℜ3.

4. Find the dot product of two vectors in ℜ2 and ℜ3.

5. Use the dot product to find the angle between two vectors in ℜ2 and ℜ3.

6. Find the projection of a vector onto another vector in ℜ2 and ℜ3, and express the original vector as a sum of two orthogonal vectors.

7. Find the distance between a point and a line in ℜ2 and ℜ3.

Rev.F09

Credit

Some of these slides have been adapted/modified in part/whole from the text or slides of the following textbooks:

• Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition

• Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition

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